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Moving Charges And Magnetism

Question
CBSEENPH12039193

Considering the case of a parallel plate capacitor being charged, show how one is required to generalize Ampere’s circuital law to include the term due to displacement current.

Solution

Ampere’s circuital law states that,
contour integral B with rightwards harpoon with barb upwards on top space. space stack d l with rightwards harpoon with barb upwards on top space equals space mu subscript o space I

Electric flux across a parallel capacitor is given by, 
straight ϕ subscript straight E space space equals space EA space equals space 1 over straight epsilon subscript straight o straight Q over straight A xA space equals space straight Q over straight epsilon subscript straight o 

Current in the plates of the capacitor is given by, 
i space equals space fraction numerator d Q over denominator d t end fraction
italic therefore italic space fraction numerator d ϕ subscript E over denominator d t end fraction italic space italic equals italic space fraction numerator d over denominator d t end fraction open parentheses Q over epsilon subscript o close parentheses italic space italic equals italic space italic 1 over epsilon subscript o fraction numerator d Q over denominator d t end fraction

italic rightwards double arrow italic space fraction numerator d ϕ subscript E over denominator d t end fraction italic space italic equals italic space fraction numerator d Q over denominator d t end fraction italic equals i
This is the missing term in the ampere's law. 

Therefore, total current in the conductor is the sum of displacement current and conduction current.
straight i space equals space straight i subscript straight c space plus space straight i subscript straight d space equals space space straight i subscript straight c space plus space straight epsilon subscript straight o space dϕ over dt

Putting space the space value space of space straight I space in space Amper apostrophe straight s space Circuital space law comma space we space have

contour integral straight B with rightwards harpoon with barb upwards on top. dl with rightwards harpoon with barb upwards on top space equals space straight mu subscript straight o space straight I subscript straight c space plus space straight mu subscript straight o element of subscript straight o space dϕ subscript straight E over dt

This space is space the space required space generalized space form space of space Ampere ’ straight s space law.